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We prove that the integral closure of a strongly Golod ideal in a polynomial ring over a field of characteristic zero is strongly Golod, positively answering a question of Huneke. More generally, the rational power
of an arbitrary homogeneous ideal is strongly Golod for
is strongly Golod, then
is strongly Golod for
. We also show that all the coefficient ideals of a strongly Golod ideal are strongly Golod.
An inequality is established involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, a proof is given of a result due to Goto and Nakamura (first conjectured by Watanabe and Yoshida), which states that the Hilbert–Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal. The result is also further refined.
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